**Significance Statement**

Viscoelasticity can be exploited to synthesize materials with exceptional damping properties. Notably polymer composites have been designed for damping and noise reduction in automobiles and airplanes, conferring structural stability during wind and earthquake induced vibrations and mechanical damping in number of applications. Further work on viscoelasticity in crystalline materials such as metals and alloys by researchers mainly focused on maintaining high stiffness (high elastic modulus) while providing significant damping.

Researchers led by Professor Keblinski from Department of Material Science and Engineering at Rensselaer Polytechnic Institute (USA) used non-equilibrium molecular dynamics shear simulations to study viscoelasticity of model crystalline composites with various microstructures as function of shear rate. Their study is published in journal, *Composites Part B*.

Theoretical study in predicting bounds for elastic and shear moduli for materials are either based on well-known variational theory (*Hashin and Shtrikman, Mech Phys Solids;1963*) or couple homogenization theory to obtain bounds for materials properties and a structural optimization algorithm to model microstructures with tunable properties. Despite all these valuable predictive tools, a deeper understanding of underlying mechanisms is possible only using molecular level simulations.

For implementation of simulation, model composites comprised of a stiff spherical inclusions inside a soft matrix. Both phases are crystalline in a face centered cubic lattice and are modeled by the standard of Lennard-Jones potential.

Viscoelastic properties were studied using non-equilibrium oscillatory shear deformation simulations and an equation having homogenous, sinusoidal shear strain γ_{xy} with a shear frequency (*f*) to the cubic simulation cell is applied. Shear stress τ_{xy} derived was now described by a sinusoidal function having a phase shift angle (ø). Accumulation of shear stress τ_{xy} over multiple shear cycles was used to calculate shear modulus (G) which is the ratio of maximum shear stress to maximum shear strain.

System was equilibrated at 40K and zero pressure using a Nose-Hoover thermostat for all shear simulations and damping time constants for thermostats and barostats are 100fs and 200fs respectively which was then followed by further equilibrium at constant volume and constant temperature ensemble for 400ps. Equilibrated structures were then subjected to oscillatory shear simulations at constant volume and constant energy ensemble to characterize viscoelasticity.

Results for vibrational analysis using Gruneisen parameter *γ*_{1} which indicated energy dissipation due to anharmonic coupling showed that at any variation *γ*_{1 }for various composites ranging from ø=0 (fully soft) to ø=1 (fully stiff) for L=7 unit cells, ø=0 shows a maximum *γ*_{ }of about 8 at low frequency end (0.2THz) while pure stiff phase (ø=1) had a maximum *γ* of about 5. Two intermediate volume fraction (ø=0.27 and ø=0.47) exhibits much larger. Soft phase was seen to experience much larger shear strain compared to stiff phase. Hence, confinement of a softer phase in a composite can thus potentially result in significant viscous damping via large local transformation.

Effect of inclusion volume fraction under fixed simulation box size of L=7 and constant shear frequency of 0.33THz showed loss modulus was close to zero for homogenous composition (ø=0 and ø=1). At intermediate fractions of stiff phase (0.2<ø_{diff}<0.6) there is a significant viscoelastic damping and maximum loss modulus of -1500MPa which is approximately 25 times larger than homogenous soft phase. It was also seen that as contrast between phases become bigger, the greater is the strain accommodated by softer phase. Loss modulus increases almost linearly with relative stiffness (_{rel}) in range of 2-4 and begins to saturate at ε_{rel}>5.

With ø=5 and three different structures of single inclusion with L=7, 14 and 21 unit cells, result of frequency-sweep simulations showed maximum shear strain fixed at 1.5% for all cases. Loss modulus was negligible small at low frequency of *f∼*0.01THz and high frequency of *f∼*0.02THz. At intermediate frequencies (0.3<*f*<1.5THz) shows large loss modulus. Thus, damping is greatly influenced for composites near shear frequency while there are no artifacts from choice of simulation cell size used since identical results for three cases was observed.

Superlattice structures were seen to exhibit similar damping behavior. With superlattice feature (width of one layer of soft and stiff phase) varied from L=4 unit cells (10.8A) to L=24 unit cells (64.5A) showed high loss moduli at intermediate frequencies. As superlattice period decreases, the low frequency peak intensity decreases and high frequency peak increases.

This clearly shows that low frequency peak originates from the layering in superlattice which disappears as superlattice period diminishes. Variation of two peak frequencies in function of inverse superlattice width also showed that high-frequency peak has a slope of 3533.8m/s which is about 1.4 times larger than that of low frequency peak.

**Journal Reference **

Raghavan Ranganathan, Rahmi Ozisik, Pawel Keblinski.** ****Viscoelastic Damping in Crystalline Composites: A Molecular Dynamics Study. ** Composites Part B: Engineering, Volume 93, 2016, Pages 273–279.

Department of Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA.

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